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I have a probability vector $p$ s.t. $1^Tp=1$ and $p\geq 0$. I want to compute the spectral radius of the matrix $M=\text{diag}(p)-pp^T$ where $\text{diag}(p)$ has its diagonal elements as $p$ and its off-diagonals as 0.

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A more general problem is addressed in section 5 of Golub, G. H. "Some modified matrix eigenvalue problems". SIAM Review 15, 318 (1973).

For background and other references see the Wikipedia article on the Bunch-Nielsen-Sorensen formula.

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